World Scientific
  • Search
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries. by:1 (Source: Crossref)

Using an inequality due to Ricard and Xu, we give a different proof of Paul Skoufranis’s recent result showing that the strong convergence of possibly non-commutative random variables X(k)X is stable under reduced free product with a fixed non-commutative random variable Y. In fact we obtain a more general fact: assuming that the families X(k)={Xi(k)} and Y(k)={Yj(k)} are ∗-free as well as their limits (in moments) X={Xi} and Y={Yj}, the strong convergences X(k)X and Y(k)Y imply that of {X(k),Y(k)} to {X,Y}. Phrased in more striking language: the reduced free product is “continuous” with respect to strong convergence. The analogue for weak convergence (i.e. convergence of all moments) is obvious. Our approach extends to the amalgamated free product, left open by Skoufranis.

Communicated by L. Accardi

AMSC: 46L54, 46L07


  • 1. G. Anderson, Convergence of the largest singular value of a polynomial in independent Wigner matrices, Ann. Probab. 41 (2013) 2103–2181. Crossref, Web of ScienceGoogle Scholar
  • 2. Z. D. Bai and J. W. Silverstein, No eigenvalues outside the support of the limiting spectral distribution of large dimensional sample matrices, Ann. Probab. 26 (1998) 316–345. Crossref, Web of ScienceGoogle Scholar
  • 3. D. Blecher and C. Le Merdy, Operator algebras and their modules — an operator space approach (Oxford Univ. Press, 2004). CrossrefGoogle Scholar
  • 4. N. P. Brown and N. Ozawa, C∗-algebras and Finite-Dimensional Approximations (Amer. Math. Soc., 2008). CrossrefGoogle Scholar
  • 5. B. Collins and C. Male, The strong asymptotic freeness of Haar and deterministic matrices, Ann. Sci. Éc. Norm. Supér. 47 (2014) 147–163. Crossref, Web of ScienceGoogle Scholar
  • 6. K. Dykema, Faithfulness of free product states, J. Funct. Anal. 154 (1998) 323–329. Crossref, Web of ScienceGoogle Scholar
  • 7. U. Haagerup and S. Thorbjørnsen, A new application of random matrices: Ext(Cred(𝔽2)) is not a group, Ann. Math. 162 (2005) 711–775. Crossref, Web of ScienceGoogle Scholar
  • 8. U. Haagerup, H. Schultz and S. Thorbjørnsen, A random matrix approach to the lack of projections in Cred(𝔽2), Adv. Math. 204 (2006) 1–83. Crossref, Web of ScienceGoogle Scholar
  • 9. S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980) 72–104. Web of ScienceGoogle Scholar
  • 10. E. C. Lance, Hilbert C∗-Modules. A Toolkit for Operator Algebraists (Cambridge Univ. Press, 1995). CrossrefGoogle Scholar
  • 11. C. Male, The norm of polynomials in large random and deterministic matrices with an appendix by Dimitri Shlyakhtenko, Probab. Theory Relat. Fields 154 (2012) 477–532. Crossref, Web of ScienceGoogle Scholar
  • 12. D. Paul and J. W. Silverstein, No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix, J. Multivariate Anal. 100 (2009) 37–57. Crossref, Web of ScienceGoogle Scholar
  • 13. G. Pisier, A simple proof of a theorem of Kirchberg and related results on C-norms, J. Op. Theory. 35 (1996) 317–335. Google Scholar
  • 14. E. Ricard and Q. Xu, Khintchine type inequalities for reduced free products, and applications, J. Reine Angew. Math. 599 (2006) 27–59. Web of ScienceGoogle Scholar
  • 15. H. Schultz, Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases, Probab. Theory Relat. Fields 131 (2005) 261–309. Crossref, Web of ScienceGoogle Scholar
  • 16. P. Skoufranis, On a notion of exactness for reduced free products of C-algebras, J. Reine Angew. Math. 700 (2015) 129–153. Crossref, Web of ScienceGoogle Scholar
  • 17. D. V. Voiculescu, K. Dykema and A. Nica, Free Random Variables (Amer. Math. Soc., 1992). CrossrefGoogle Scholar
Remember to check out the Most Cited Articles!

Check out Probability & Statistics books in our Mathematics 2021 catalogue
Featuring authors Ingram Olkin, Takeyuki Hida and more.